Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Algebră şi Geometrie (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MG264	Critical Points and Applications	

Semester	Hours: C+S+L	Category	Status	
1	2+1+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
PINTEA Cornel	Ph.D.	Assoc.Prof.	Geometry	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives

     The course is studing on the one hand the local behaviour of a real differentiable function around a non-degenerate critical point and the Morse Theory as an immediate application, and on the other hand is studing the critical set of mappings with values in manifolds of dimension greater than one.

IV. Course contents

I. ELEMENTS OF CRITICAL POINT THEORY
   1.1 Generalities on Critical/Regular Points and Critical/Regular Values
   1.2 The first Deformation Lemma
   1.3 The second deformation Lemma

II. LUSTERNICK-SCHNIRELMANN THEORY
   2.1 Elements of Lusternick-Scnirelmann category. 
   2.2 Properties of the Lusternick-Schnirelmann category. Minimax principle.
   2.2  The Lusternick-Scnirelmann theorem.

III. MORSE THEORY AND APPLICATIONS
   3.1 The Morse lemma
   3.2 Passing a critical level
   3.3 The Morse inequalities

IV CRITICAL POINTS OF CLOSED ONE FORMS

    4.1 Morse type zeros of closed 1-forms
    4.2 Weakly complete closed 1-forms
    4.3  Closed 1-forms on closed connected manifolds having at most one zero

V. Bibliography

1. Biasi, C., Saeki, O.,  On transversality with deficiency and a conjecture of Sard, Trans. AM.S. Vol. 350,1996,  pp. 5111-5112.
2. Burghelea, D., Hangan, Th., Moscovici, H., Verona, A., Introducere in topologia diferentiala, Ed. Stiintifica, Bucuresti, 1973.
3. Farber, M., Schutz, D., Closed 1-forms with at most one zero, Topology, 45 (2006) pp. 465–473.
4. Milnor, J., Morse Theory, Annals of Math. Studies, Princeton Univ. Press, 1963.
5. Palais, R., S., Terng, C-L., Critical Point Theory and Submanifold Geometry, Lectures Notes in Mathematics, Springer-Verlag.
6. Pintea, C., Continuous Mappings with an Infinite Number of Topologically Critical Points, Annales Polonici Mathematici, LXVII.1, 1997.

VI. Thematic of didactic activities per weeks

Săptămâna 1 Differential manifolds and differentiable maps

Săptămâna 2 Critical points and critical values. The rank theorem

Săptămâna 3 The Sard theorem

Săptămâna 4 The first deformation theorem

Săptămâna 5 A doua teoremă de deformare. Principiul gulerului necritic 

Săptămâna 6 Elements of Lusternick-Scnirelmann category. The minimax principle

Săptămâna 7. The Lusternick-Scnirelmann theorem

Săptămâna 8 The Morse lemma

Săptămâna 9 Passing a critical level

Săptămâna 10 Chain and cochain complexes 

Săptămâna 11 Singular homology and singular cohomology

Săptămâna 12 The Morse inequalities

Săptămânile 13-14 Zeros of closed 1-forms

VII. Didactic methods used

Exposition, Conversation, Problematization

VIII. Assessment

Tutorial activity(20%)+Reports(30%)+Exam(50%).

IX. Additional references

	Dean,							Chair head,
            Prof. Ţâmbulea Leon, Ph.D.                                    Prof.  Andrica Dorin, Ph.D.

	Date,							Course responsible,
								Assoc.Prof. PINTEA Cornel, Ph.D.